Half-angle identities determine squared values first; the quadrant supplies the sign.
WHAT HAPPENS IF…At θ=240°, what sign should cos(θ/2) have?
Interactive exploration
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Connect the picture to the ideasin2(θ/2)=21−cosθ,cos2(θ/2)=21+cosθ
Substitute θ/2 into the double-angle identities and solve for the squares. Taking a square root gives a magnitude, not a sign. In the displayed interval θ/2 lies in [0,π], so sine is nonnegative while cosine becomes negative beyond θ=π.
A detail that matters
Do not attach the positive square root automatically to every half-angle value.